The **331 model** in particle physics offers an explanation of why there must exist three families of quarks and leptons. One curious feature of the Standard Model is that the anomaly cancels exactly, for each quark-lepton family, of which we know three. The standard model thus offers no explanation of why there are three families, or indeed why there is more than one family.

One idea, therefore, is to extend the standard model such as to destroy the perfect cancellation of the anomaly, per family, and to make the three families transform differently under an extended gauge group, and to arrange that the anomaly cancel, only for three families. But the cancellation will persist for 6, 9, ... families, so then there is a new super-family problem, which is best avoided by having only three families.

Such a construction necessarily requires the addition of further gauge bosons and chiral fermions, which then provide testable predictions of the new model, in the form of elementary particles, to be sought experimentally, at masses above the weak scale, of about 100 GeV. The minimal 331 model predicts singly and doubly charged spin-one bosons, bileptons, which could show up in electron-electron scattering when it is studied at TeV energy scales and may also be produced in multi-TeV protonâ€“proton scattering at the Large Hadron Collider as early as 2011.

The 331 model offers an explanation of why there must exist three families of quarks and leptons, a fact which is put in "by hand" in the Standard Model. The 331 model is an extension of the electroweak gauge symmetry from to with and the hypercharge and the electric charge where T_{3} and T_{8} are the Gell-Mann matrices of SU(3)_{L} and Î˛ and I are parameters of the model. The name 331 comes from the full gauge symmetry group .

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—Susan Sontag (b. 1933)